A trip to the consulate

While the U.S. is (generally) cool with dual citizenship, South Korea is not. When my wife naturalized in 2017, she had to go to the Korean consulate in Honolulu to de-register.

Our kids, however, did not -- they don't have to choose which one to keep until they turn 18.

There's a catch, though. Once a young man turns 18, he can't de-register unless he's served his mandatory [unpaid] military service, and a teenager who waits too long puts himself at risk of being drafted. This has happened to Korean-Americans who gone to Korea to teach English.
So today, my kids surrendered their South Korean citizenship at the consulate in Frankfurt. Not just my son, but my daughter as well -- given the country's collapsing birth rate and lingering security risks (looking at you, NK) the SK government has considered expanding the draft to include women.

If in the future my kids want to live or work in Korea, they can apply for an F-4 visa (same as my wife) but as of today, they'll have to use the "foreigner" line at the airport.